# Response Surface Model by 6kneY7

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```									Response Surface Model

NASCAR TRACK Top Qualifying
Speeds (Through 2009)
Data Description
• Units: NASCAR Sprint Cup Tracks
• Response Variable: Y=Top Qualifying
Speed for NASCAR Sprint Cup Series
• Predictor Variables:
X1 = Bank of Turns (Degrees)
X2 = Length of Track (Miles)
• Hypotheses:
As Bank Top Speed
As Track Length Top Speed
Response Surface Model
Second Order Polynomial With Interaction
Yi   0  1 X i1   2 X i 2  11 X i2  12 X i1 X i 2   22 X i22   i
1

SurfaceDerivatives wrt X 1 , X 2 :
E (Y )                                                E (Y )
 1  211 X i1  12 X i 2                             2  12 X i1  2 22 X i 2
X 1                                                   X 2
Centered Model (Reduces Collineari ty Among Predictors) :
              
Yi   0   1 X i1  X .1   2 X i 2  X .2   11 X i1  X .1                     2

 12X  X X  X   X  X   
i1         .1        i2        .2    22        i2     .2
2
i

    X   X   X    X X                                                   
2                                      2
0      0     1         .1        2   .2    11    .1        12         .1   .2     22 X .2
1   1  2 11 X .1   12 X .2  2   2   12 X .1  2 22 X .2
11   11 12   12  22   22
Data
Track                                  Bank     Length   TopQual    Cbank    Clength    CBank2 Clength2   CBL
ATLANTA MOTOR SPEEDWAY                  24       1.54    197.478    4.690      0.018    22.0006 0.0003  0.0849
BRISTOL MOTOR SPEEDWAY                  36       0.53    128.709    16.690    -0.992   278.5720 0.9839 -16.5554
CALIFORNIA SPEEDWAY                     14         2     187.432    -5.310     0.478    28.1910 0.2286  -2.5385
CHICAGOLAND SPEEDWAY                    18        1.5    188.147    -1.310    -0.022    1.7149  0.0005  0.0287
DARLINGTON RACEWAY                      24       1.37    173.797    4.690     -0.152    22.0006 0.0231  -0.7125
DAYTONA INTERNATIONAL SPEEDWAY          31        2.5    210.364    11.690     0.978   136.6672 0.9567  11.4344
DOVER INTERNATIONAL SPEEDWAY            24         1     161.522    4.690     -0.522    22.0006 0.2724  -2.4480
HOMESTEAD-MIAMI SPEEDWAY                19        1.5    181.111    -0.310    -0.022    0.0958  0.0005  0.0068
INDIANAPOLIS MOTOR SPEEDWAY              9        2.5    186.293   -10.310     0.978   106.2863 0.9567 -10.0837
KANSAS SPEEDWAY                         15        1.5    176.449    -4.310    -0.022    18.5720 0.0005  0.0944
LAS VEGAS MOTOR SPEEDWAY                20        1.5    172.563    0.690     -0.022    0.4768  0.0005  -0.0151
LOWES MOTOR SPEEDWAY                    24        1.5    186.034    4.690     -0.022    22.0006 0.0005  -0.1027
MARTINSVILLE SPEEDWAY                   12       0.53     95.371    -7.310    -0.992    53.4291 0.9839  7.2504
MICHIGAN INTERNATIONAL SPEEDWAY         18         2     194.232    -1.310     0.478    1.7149  0.2286  -0.6261
NEW HAMPSHIRE INTERNATIONAL SPEEDWAY    4.5      1.06    132.089   -14.810    -0.462   219.3220 0.2134  6.8406
NORTH CAROLINA SPEEDWAY                 23       1.02    158.035    3.690     -0.502    13.6196 0.2519  -1.8523
PHOENIX INTERNATIONAL RACEWAY           10         1     134.718    -9.310    -0.522    86.6672 0.2724  4.8587
POCONO RACEWAY                           9        2.5    172.391   -10.310     0.978   106.2863 0.9567 -10.0837
RICHMOND INTERNATIONAL RACEWAY          14       0.75    126.499    -5.310    -0.772    28.1910 0.5958  4.0984
TALLADEGA SUPERSPEEDWAY                 33       2.66    212.809    13.690     1.138   187.4291 1.2953  15.5811
TEXAS MOTOR SPEEDWAY                    24        1.5    192.137    4.690     -0.022    22.0006 0.0005  -0.1027

Mean                                   19.310   1.522    169.913   0.000      0.000

Variables beginning with “C” are centered
Second-Order Polynomial Fit
SUMMARY OUTPUT

Regression Statistics
Multiple R                                        0.9877
R Square                                          0.9755
Standard Error                                    5.5244
Observations                                          21

ANOVA
df            SS          MS         F       Significance F
Regression                                       5         18247.55     3649.51    119.58        0.0000
Residual                                        15          457.79       30.52
Total                                           20         18705.34

Coefficients Standard Error   t Stat   P-value    Lower 95% Upper 95%
Intercept                                    182.8680        1.7063     107.1743    0.0000     179.2312 186.5048
Cbank                                           1.4743       0.1516      9.7263     0.0000      1.1512    1.7974
Clength                                       43.8591        2.0541      21.3523    0.0000      39.4809   48.2372
CBank2                                         -0.0141       0.0215      -0.6522    0.5241      -0.0600   0.0319
Clength2                                      -30.7613       4.0328      -7.6277    0.0000     -39.3571  -22.1656
CBL                                             0.0455       0.1762      0.2583     0.7997      -0.3300   0.4210

Note: Cbank, Clength, and Clength2 are highly significant. Neither the interaction
term (CBL) or the quadratic term in Bank (CBank2) appear to be significant
Reduced Model
SUMMARY OUTPUT

Regression Statistics
Multiple R                                           0.9873
R Square                                             0.9747
Standard Error                                       5.2794
Observations                                             21

ANOVA
df             SS            MS           F       Significance F
Regression                                          3          18231.51       6077.17      218.04        0.0000
Residual                                           17           473.83         27.87
Total                                              20          18705.34

Coefficients Standard Error   t Stat       P-value     Lower 95% Upper 95%
Intercept                                       182.6469        1.5972     114.3549        0.0000      179.2771 186.0167
Cbank                                             1.4639        0.1437      10.1854        0.0000       1.1607    1.7672
Clength                                          44.1766        1.9178      23.0354        0.0000       40.1305   48.2228
Clength2                                         -32.5218       2.8254     -11.5107        0.0000      -38.4828  -26.5608

H 0 :  12   22  0 H A :  12 and/or  22  0
 SS RESRe duced   SS RESFULL       473.83  457.79 
                                      
 df RES (Re duced)  df RES ( Full )        17  15     8.02
T .S .                                                               0.26 F.05, 2,15  3.682 PF2,15  0.263  0.772
 SS RESFULL                       457.79       30.52
                                    15 
         
 df RES ( Full ) 
                  
Plot of Reduced Model (Original Units)
250

200

150

Bank10
Bank20
Bank30
100

50

0
0    0.5   1    1.5   2    2.5   3
3-D Surface Plot

250

200-250
200
150-200
100-150
150   50-100
0-50

5
100
9
13
17
50
21
25
29
0
33
2.7 2.9
37                                         2.3 2.5
1.7 1.9 2.1
1.3 1.5
0.9 1.1
0.5 0.7
Contour Plot
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2
1.9
1.8
200-250
1.7
150-200
1.6
100-150
1.5   50-100
1.4   0-50
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
3   4   5   6   7   8   9   10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

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